Willing? Huh, my druid is perfectly willing to wear breastplate and not die.
No he's not, not under the standard rules. Druids will not wear metal armour, so if your character is a druid, they will not wear metal armour. That' the rules.
So, why is it that I have to go on a quest to earn the right to wear it?
You don't have to. You can wear hide.
And so might a metal one. That isn't the point. Why do I have to go on a quest to get non-magical breastplate? No one else to. And the designers clearly stated it isn't a balance issue for me to have non-magical breastplate, so why the hoops?
Because its part of the class that they don't wear metal armour.
But you said there was no non-metal breastplates in the PHB.
There aren't. GM can make such up, but they're not required to do so. But if the GM lets wizards to buy spell scrolls, then it means that magic items are in the game and buyable, so druid should also be able to buy a low-level magic armour with unusual material quality.
And sure, better armor is never required. You can have everyone play with no armor if you really wanted too... but it is kind of assumed, at some point, that you are going to improve your gear. And yet, while the cleric of nature who worships Sylvanus can buy that metal breastplate in window, Sylvanus declared that druids must never wear metal armor, so the Druid who worships him can't. Because reasons that are reasons.
Yes. Because druids are not clerics. But if that bothers you, you can certainly play that nature cleric instead, and buy that metal armour. See, almost like these classes were meaningfully different from each other!
Nothing on page 66 refers to going on quest to see more animals. I thought you refering to the rules in Xanathars on pg 24. Which are optional.
It says you can turn into animals you've seen. You presumably need to go into places where these animals live in order to see them.
But it isn't a weakness. Druids can wear the same exact armors as clerics. You are just having them jump through hoops for "theme" because metal taken from the earth is somehow unnatural, but only when shaped into armor and shields.
Sounds like a weakness to me.
Heck, there is literally, literally no reason a druid should not be able to use a metal shield. They provide the exact same AC bonus, they aren't being "worn" and yet Druids can't use metal shields because that would be naughty and they are supposed to have a "weakness" in that?
That's not really a weakness, as nonmetal shields are just as good. The armour limitation however is.
And I would have no patience for being forced to jump through hoops for mundane gear that just has a different coat of paint on it.
Then don't play a druid. Or any character that requires expensive gear for that matter.
And if it is so interesting and so fun to do, why not require everyone to do it?
We generally do. That gold to the paladin's full plate must come from somewhere. Whether you adventure and find gold and buy gear with that, or whether you adventure and gather animal bits and make armour out of those or whether you adventure and find magical items it is basically the same thing. You adventure in order to gain better gear. This is literally the classic premise of large part of the game.